Question: Teams $ A$, $ B$, and $ C$ are playing a game of strength. Each team has attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below). Team $ A$ pulls with force vector ${\vec{a}} = 4\hat{i} + 0\hat{j}$ Team $ B$ pulls with force vector ${\vec{b}} = -2\hat{i} + 4\hat{j}$ Team $ C$ pulls with force vector ${\vec{c}} = -3\hat{i} - 3\hat{j}$ (Forces are given in kilo newtons, $\text{kN}$.) Which team will win the game? Choose 1 answer: Choose 1 answer: (Choice A) A Team $ A$ (Choice B) B Team $ B$ (Choice C) C Team $ C$ What is the magnitude of the teams' combined force acting on the ring?
Answer: Whenever forces are pulling in different directions, they tend to partially cancel each other out. This canceling effect is exactly what happens when we add vectors. Consider what happens when we add the three vectors presented in the problem. $\begin{aligned} {\vec{a}} + {\vec{b}} + {\vec{c}} &= ( 4 \hat i + 0\hat j ) + ( -2 \hat i + 4\hat j ) + ( -3 \hat i - 3\hat j ) \\\\ &=(4+ (-2) + (-3)) \hat i + (0+4-3) \hat j \\\\ &=-1\hat i + 1\hat j \end{aligned}$ The vector $\vec n = -1\hat i + 1\hat j$ describes the net force acting on the ring. From this, we can already see that Team $ B$ will win. We can find the magnitude of $ \vec n $ using the Pythagorean theorem. $\begin{aligned} \| \vec n \|^2 &= -1^2 + 1^2\\\\ \| \vec n \| &= \sqrt{-1^2 + 1^2}\\\\ \| \vec n \| &= \sqrt{2}\\\\ \| \vec n \| &\approx 1.4 \text{ kN} \end{aligned}$ Finding the direction of $\vec n$ will tell us in what direction the ring is getting pulled. $\vec n$ is pointing in the second quadrant with an $x$ -component of $-1$ and a $y$ -component of $1$. We can find the direction of any vector $\vec v$ in the second quadrant using the arctangent function and adding $\pi$. $\begin{aligned} \tan \theta &= \dfrac{ y}{ x}\\\\ \tan \theta &= \dfrac{1}{-1}\\\\ \theta &= \arctan{(-1)} \\\\ \theta&\approx -0.785 \text{ rad} \end{aligned}$ Adding $\pi$ to this result gives us the actual direction, $2.4$ radians (rounded to the nearest tenth). Team $ B$ will win. The magnitude of the combined force acting on the ring is $1.4 \text{ kN}$. The direction the ring is getting pulled is $2.4$ radians.